The creation of complex numbers is found in the exploration of square roots of negative numbers, a notion that seemed incongruous within Euclid’s axioms and then present rules governing integers. The problem presented by the square root of negative numbers spurred a significant shift in thinking, leading to the conceptualization and acceptance of “imaginary” numbers, as termed by René Descartes.

The Italian mathematician Girolamo Cardano, in the 16th century, was among the first to acknowledge that while the square root of a negative number may not reside within the realm of real numbers, it could indeed possess an “imaginary” essence. This realization paved the way for Rafael Bombelli, who meticulously outlined how equations could be solved using complex numbers, thereby introducing a “real” component alongside an “imaginary” component based on the unit i, thanks to Leonhard Euler’s notation.

Complex numbers are expressed as a combination of these two components, for instance, 3+2i. The introduction of the complex plane by Jean-Robert Argand enriched the understanding of complex numbers, offering a graphical representation that plots real and imaginary components on perpendicular axes. This innovative approach demystified complex numbers and also laid the groundwork for advanced mathematical constructs like quaternions, introduced by William Rowan Hamilton, which extend complex numbers into a four-dimensional space.